for all sequences \(\lbrace h_{n}\rbrace \subset X\)  and \(\lbrace t_{n}\rbrace \subset \mathbb {R}_{+}\) 
such that \(t_{n}\downarrow 0\)  and \(h_{n}\rightarrow h\in X_{0}\)  as \(n\rightarrow \infty \) 
and \(x+t_{n}h_{n}\in X_{D}\)  for all \(n\) . Two related differentiability
concepts are defined as follows. If the convergence (REF )
is required to hold only for all fixed \(h_{n}\equiv h\in X_{0}\) ,
the map \(\phi \)  is said to be Gateaux directionally differentiable
(at \(x\in X_{D}\)  tangentially to \(X_{0}\) ). Alternatively, if in
the above definition the requirement “\(\lbrace t_{n}\rbrace \subset \mathbb {R}_{+}\) 
such that \(t_{n}\downarrow 0\) ” is replaced with “\(\lbrace t_{n}\rbrace \subset \mathbb {R}\) 
such that \(t_{n}\rightarrow 0\) ”, the map is said to be (fully) Hadamard
differentiable (at \(x\in X_{D}\)  tangentially to \(X_{0}\) ). In statistical
literature, these differentiability concepts are discussed and developed
in [1]}, [2]}, and Shapiro
[3]}, [4]}, among others.
